Complexity in algebraic QFT

TL;DR

This paper introduces a new measure of quantum channel complexity in relativistic quantum field theory based on the Belavkin-Staszewski divergence, demonstrating its key properties and applications to QFT structures.

Contribution

It defines a novel complexity measure for quantum channels in QFT using BS divergence and proves its fundamental properties and specific values for certain channels.

Findings

Complexity is subadditive for composite channels.

Complexity is additive for spacelike separated channels.

For an N-ary measurement, complexity equals log N.

Abstract

We consider a notion of complexity of quantum channels in relativistic continuum quantum field theory (QFT) defined by the distance to the trivial (identity) channel. Our distance measure is based on a specific divergence between quantum channels derived from the Belavkin-Staszewski (BS) divergence. We prove in the prerequisite generality necessary for the algebras in QFT that the corresponding complexity has several reasonable properties: (i) the complexity of a composite channel is not larger than the sum of its parts, (ii) it is additive for channels localized in spacelike separated regions, (iii) it is convex, (iv) for an $N$-ary measurement channel it is $\log N$, (v) for a conditional expectation associated with an inclusion of QFTs with finite Jones index it is given by $\log (\text{Jones Index})$. The main technical tool in our work is a new variational principle for the BS divergence.